TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-09 (gpt-6-astra)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Alexandrov's theorem, under Dependent Choice: a subspace of a complete metric space is completely metrizable exactly when it is $G_{δ}$

Statement

Assume Dependent Choice. For a subspace $Y$ of a complete metric space $X$ , $Y$ is completely metrizable if and only if $Y$ is a $G_{δ}$ subset of $X$ .

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

Under Countable Choice, if $(X, d)$ is complete and $Y \subseteq X$ is $G_{δ}$ in $X$ , then $Y$ is completely metrizable. (Under the Axiom of Countable Choice, every $G_{δ}$ subspace of a complete metric space is completely metrizable)

[F2]

Assume Dependent Choice. If $Y$ is a completely metrizable subspace of a metric space $X$ , then $Y$ is a $G_{δ}$ subset of $X$ . (Under Dependent Choice, every completely metrizable subspace of a metric space is $G_{δ}$ ).

[F3]

DC supplies a sequence beginning at any specified point of an entire relation; Countable Choice selects from any given sequence of nonempty sets. (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

technique · direct

1.1givenF1F2

The empty subspace satisfies both conditions.

1.2

The assumed DC supplies the Countable Choice needed by [F1]. [given, F3] For any sequence of nonempty sets, take the set of finite lists choosing from its first finitely many members. This set contains the empty list, and the one-term-extension relation is entire. DC starting at the empty list gives nested lists of every finite length; their union is the required choice function. No implication theorem from a later choice page is used.

2.1

Apply [F1] with the given complete ambient metric and the Countable [step 1.2, F1, F2] Choice just derived. For the converse apply [F2] under the given DC; it needs only a compatible complete metric on $Y$ , not completeness of its inherited metric. Thus both implications have their stated hypotheses.

3.1step 2.1∎

The preceding construction and implications establish the assertion.

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)
Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)

Alexandrov's theorem, under Dependent Choice: a subspace of a complete metric space is completely metrizable exactly when it is GδGδ​